Asymptotic resemblance relations on groups (Q2078107)

The canonical coarse structure on groups is generalized in this paper to compatible coarse structures on groups. Now, every coarse structure induces an asymptotic resemblance relation on subsets of the space. In particular, asymptotic resemblance on groups induced by compatible coarse structures are discussed. Asymptotic disjointness is another relation on subsets of the space. It satisfies the separation property asymptotically normal if the coarse space is metrizable. This paper provides an example for a group which is not asymptotically normal. Asymptotic dimensiongrad is an invariant on coarse spaces very similar to asymptotic dimension and can also be defined on groups with compatible coarse structures. The author defines a set theoretic coupling for groups with asymptotic resemblance relations induced by compatible coarse structures. The author shows that if two groups admit a set theoretic coupling then they are asymptotically equivalent.